Questions — Edexcel C2 (579 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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Edexcel C2 Q9
13 marks Moderate -0.3
  1. Solve, for \(0° < x < 180°\), the equation \(\sin (2x + 50°) = 0.6\), giving your answers to 1 d. p. [7]
  2. In the triangle \(ABC\), \(AC = 18\) cm, \(\angle ABC = 60°\) and \(\sin A = \frac{1}{3}\).
    1. Use the sine rule to show that \(BC = 4\sqrt{3}\). [4]
    2. Find the exact value of \(\cos A\). [2]
Edexcel C2 Q1
4 marks Moderate -0.8
The point \(A\) has coordinates \((2, 5)\) and the point \(B\) has coordinates \((-2, 8)\). Find, in cartesian form, an equation of the circle with diameter \(AB\). [4]
Edexcel C2 Q2
6 marks Moderate -0.8
\includegraphics{figure_1} The circle \(C\), with centre \((a, b)\) and radius \(5\), touches the \(x\)-axis at \((4, 0)\), as shown in Fig. 1.
  1. Write down the value of \(a\) and the value of \(b\). [1]
  2. Find a cartesian equation of \(C\). [2]
A tangent to the circle, drawn from the point \(P(8, 17)\), touches the circle at \(T\).
  1. Find, to 3 significant figures, the length of \(PT\). [3]
Edexcel C2 Q3
7 marks Standard +0.3
\(\text{f}(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.
  1. Given that f\((n)\) has a remainder of \(3\) when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
  2. Show that f\((n)\) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
  3. Hence show that f\((n)\) is divisible by \(3\) for all positive integer values of \(n\). [2]
Edexcel C2 Q4
8 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
  2. Write down the coordinates of the points at which the graph meets the axes. [3]
  3. Solve, for \(0 \leq x < 360°\), the equation \(\sin (x + 30°) = -\frac{1}{2}\). [3]
Edexcel C2 Q5
7 marks Moderate -0.3
The expansion of \((2 - px)^6\) in ascending powers of \(x\), as far as the term in \(x^2\), is $$64 + Ax + 135x^2.$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\). [7]
Edexcel C2 Q6
8 marks Standard +0.3
Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$2\cos^2\theta - \cos\theta - 1 = \sin^2\theta.$$ Give your answers to \(1\) decimal place where appropriate. [8]
Edexcel C2 Q7
8 marks Moderate -0.3
\includegraphics{figure_2} Fig. 2 shows the sector \(OAB\) of a circle of radius \(r\) cm. The area of the sector is \(15\) cm\(^2\) and \(\angle AOB = 1.5\) radians.
  1. Prove that \(r = 2\sqrt{5}\). [3]
  2. Find, in cm, the perimeter of the sector \(OAB\). [2]
The segment \(R\), shaded in Fig 1, is enclosed by the arc \(AB\) and the straight line \(AB\).
  1. Calculate, to 3 decimal places, the area of \(R\). [3]
Edexcel C2 Q8
12 marks Standard +0.3
\includegraphics{figure_3} Fig. 3 shows the line with equation \(y = 9 - x\) and the curve with equation \(y = x^2 - 2x + 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
  1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(R\) is bounded by the line and the curve.
  1. Calculate the area of \(R\). [7]
Edexcel C2 Q9
15 marks Moderate -0.3
For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find the coordinates of each of the stationary points, [5]
  3. determine the nature of each stationary point. [3]
The point \(A\), on the curve \(C\), has \(x\)-coordinate \(1\).
  1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
Edexcel C2 Q1
6 marks Moderate -0.8
  1. Show that \((x + 2)\) is a factor of \(f(x) = x^3 - 19x - 30\). [2]
  2. Factorise \(f(x)\) completely. [4]
Edexcel C2 Q2
8 marks Standard +0.3
For the binomial expansion, in descending powers of \(x\), of \(\left( x^3 - \frac{1}{2x} \right)^{12}\),
  1. find the first 4 terms, simplifying each term. [5]
  2. Find, in its simplest form, the term independent of \(x\) in this expansion. [3]
Edexcel C2 Q3
9 marks Moderate -0.8
The curve \(C\) has equation \(y = \cos \left( x + \frac{\pi}{4} \right)\), \(0 \leq x \leq 2\pi\).
  1. Sketch \(C\). [2]
  2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes. [3]
  3. Solve, for \(x\) in the interval \(0 \leq x \leq 2\pi\), \(\cos \left( x + \frac{\pi}{4} \right) = 0.5\), giving your answers in terms of \(\pi\). [4]
Edexcel C2 Q4
9 marks Moderate -0.8
Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of
  1. \(\log_2 (16x)\), [2]
  2. \(\log_2 \left( \frac{x^4}{2} \right)\). [3]
  3. Hence, or otherwise, solve \(\log_2 (16x) - \log_2 \left( \frac{x^4}{2} \right) = \frac{1}{2}\), giving your answer in its simplest surd form. [4]
Edexcel C2 Q5
10 marks Standard +0.3
  1. Given that \(3 \sin x = 8 \cos x\), find the value of \(\tan x\). [1]
  2. Find, to 1 decimal place, all the solutions of \(3 \sin x - 8 \cos x = 0\) in the interval \(0 \leq x < 360°\). [3]
  3. Find, to 1 decimal place, all the solutions of \(3 \sin^2 y - 8 \cos y = 0\) in the interval \(0 \leq y < 360°\). [6]
Edexcel C2 Q6
10 marks Standard +0.3
$$f(x) = \frac{(x^2 - 3)^2}{x^3}, \quad x \neq 0.$$
  1. Show that \(f(x) = x - 6x^{-1} + 9x^{-3}\). [2]
  2. Hence, or otherwise, differentiate \(f(x)\) with respect to \(x\). [3]
  3. Verify that the graph of \(y = f(x)\) has stationary points at \(x = \pm\sqrt{3}\). [2]
  4. Determine whether the stationary value at \(x = \sqrt{3}\) is a maximum or a minimum. [3]
Edexcel C2 Q7
11 marks Moderate -0.3
\includegraphics{figure_1} Fig. 1 shows part of the curve \(C\) with equation \(y = \frac{1}{3}x^2 - \frac{1}{4}x^3\). The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).
  1. Show that \(p = 6\). [1]
  2. Find an equation of the tangent to \(C\) at \(A\). [4]
The curve \(C\) has a maximum at the point \(P\).
  1. Find the \(x\)-coordinate of \(P\). [2]
The shaded region \(R\), in Fig. 1, is bounded by \(C\) and the \(x\)-axis.
  1. Find the area of \(R\). [4]
Edexcel C2 Q8
12 marks Standard +0.3
A geometric series is \(a + ar + ar^2 + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is \(S_n = \frac{a(1 - r^n)}{1 - r}\). [4]
The first and second terms of a geometric series \(G\) are 10 and 9 respectively.
  1. Find, to 3 significant figures, the sum of the first twenty terms of \(G\). [3]
  2. Find the sum to infinity of \(G\). [2]
Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.
  1. Find the exact value of the common ratio of this series. [3]
Edexcel C2 Q1
4 marks Moderate -0.3
Find the coefficient of \(x^2\) in the expansion of $$(1 + x)(1 - x)^6.$$ [4]
Edexcel C2 Q2
5 marks Moderate -0.8
A geometric series has common ratio \(\frac{1}{3}\). Given that the sum of the first four terms of the series is 200,
  1. find the first term of the series, [3]
  2. find the sum to infinity of the series. [2]
Edexcel C2 Q3
7 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) where $$f(x) = 4 + 5x + kx^2 - 2x^3,$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A\), \(B\) and \(C\). Given that \(A\) has coordinates \((-4, 0)\),
  1. show that \(k = -7\), [2]
  2. find the coordinates of \(B\) and \(C\). [5]
Edexcel C2 Q4
8 marks Moderate -0.8
    1. Sketch the curve \(y = \sin (x - 30)°\) for \(x\) in the interval \(-180 \leq x \leq 180\).
    2. Write down the coordinates of the turning points of the curve in this interval. [4]
  2. Find all values of \(x\) in the interval \(-180 \leq x \leq 180\) for which $$\sin (x - 30)° = 0.35,$$ giving your answers to 1 decimal place. [4]
Edexcel C2 Q5
9 marks Moderate -0.3
  1. Evaluate $$\log_3 27 - \log_3 4.$$ [4]
  2. Solve the equation $$4^x - 3(2^{x+1}) = 0.$$ [5]
Edexcel C2 Q6
9 marks Moderate -0.3
$$f(x) = 2 - x + 3x^{\frac{1}{2}}, \quad x > 0.$$
  1. Find \(f'(x)\) and \(f''(x)\). [3]
  2. Find the coordinates of the turning point of the curve \(y = f(x)\). [4]
  3. Determine whether the turning point is a maximum or minimum point. [2]
Edexcel C2 Q7
10 marks Moderate -0.3
The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.
  1. Show that \(\angle PQR = 90°\). [4]
Given that \(P\), \(Q\) and \(R\) all lie on circle \(C\),
  1. find the coordinates of the centre of \(C\), [3]
  2. show that the equation of \(C\) can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]